Exact solution of the Percus Yevick integral equation for hard spheres: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
m (Sligh ttidy + Cite format of references)
(Added section concerning the Carnahan-Starling equation of state)
Line 41: Line 41:


:<math>P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}</math>
:<math>P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}</math>
==A derivation of the Carnahan-Starling equation of state==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048    G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics  '''54''' pp. 1523-1525 (1971)] </ref>  Eq. 6) that one can arrive at the [[Carnahan-Starling equation of state]] by adding two thirds of the exact solution via the compressibility route, to one third via the pressure  route, i.e.
:<math>Z = \frac{ p V}{N k_B T} =  \frac{2}{3} \left[  \frac{(1+\eta+\eta^2)}{(1-\eta)^3}  \right] +  \frac{1}{3} \left[    \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2}  \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>
The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).


==References==
==References==

Revision as of 15:23, 5 April 2011

The exact solution for the Percus Yevick integral equation for the hard sphere model was derived by M. S. Wertheim in 1963 [1] (see also [2]), and for mixtures by Joel Lebowitz in 1964 [3].

The direct correlation function is given by (Eq. 6 of [1] )

where

and is the hard sphere diameter. The equation of state is given by (Eq. 7 of [1])

where is the inverse temperature. Everett Thiele also studied this system [4], resulting in (Eq. 23)

where (Eq. 24)

and

and

and where .

The pressure via the pressure route (Eq.s 32 and 33) is

and the compressibility route is

A derivation of the Carnahan-Starling equation of state

It is interesting to note (Ref [5] Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the exact solution via the compressibility route, to one third via the pressure route, i.e.

The reason for this seems to be a slight mystery (see discussion in Ref. [6] ).

References